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639 source-grounded records. Verification labels remain separate from source authentication and publication status.
Imported from VibeMathed under CC BY 4.0. Each record links to its registry entry and named primary source. Registry verification is preserved verbatim.
For which lattice parameters does a totally positive window function generate a Gabor frame? Gröchenig and Stöckler initiated the program in 2013; this paper gives the complete characterization, together with a Kadets-type theorem for shift-invariant spaces.
Strict convexity and real-analytic boundaries are what make this sharp: the classical Gordon-Webb-Wolpert drums are non-convex polygons, so the obvious escape routes are closed off.
Records, not resolutions: the exact values of these Ramsey numbers remain unknown.
integral unimodular lattices of rank at most 32; the general conjecture is open
Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem f…
Assuming the Unique Games Conjecture, it is NP-hard to approximate MAX-3-CUT better than the Frieze-Jerrum semidefinite program does, and similarly for Quantum MAX-CUT: the sharpness question in the Khot-Kindler-Mossel-O'Donnell line, connected to the Plurality is Stablest problem.
The near-quadratic Elekes-Ronyai expander conjecture over $\mathbb{R}$ predicts that a nonspecial polynomial expands any finite set to near-quadratic size. False: a fixed nonspecial quadratic polynomial, together with arbitrarily large finite sets of real algebraic integers, has image with a fixed power saving from quadratic size.
The Howland-Kato conjecture that every nonzero positive commutator $i[f(P),g(Q)]$ must arise from functions in appropriate Kato classes is false: $i[\arctan(P),\arctan(Q)]$ is nonzero and nonnegative.
The counterexample is an ordinary algebra concentrated in degree zero, with the strongest possible vanishing in positive degrees, so the phenomenon needs no grading or differential. Liu and Shen had already disproved the differential-graded version in December 2025 without any AI involvement; the classical case is the one that fell with a model in the loop.
Establishes that every PPT channel is eventually entanglement-breaking (finite EB index), in full generality, and bounds the index by 3 uniformly in dimension for a family strictly containing the 2-superpositive maps. The PPT-squared conjecture itself - index at most 2 - remains open; the paper presents its results as strong evidence toward the cubed version.
A record lower bound; the kissing number in dimension 19 remains unknown.
Does two-terminal reliability, the probability that $s$ still reaches $t$ when edges fail independently, admit a fully polynomial-time randomised approximation scheme? Asked explicitly in Kannan's 1994 survey and left open while the all-terminal cases were settled by Karger and by Guo and Jerrum. Answered positively for general graphs, both directed and undirected. The complementary unreliability question is shown…
Does bipartite bound information exist: classical correlations between two parties and an eavesdropper that cost secret bits to create, yet from which no secret key can ever be distilled?
Are the Kazhdan-Lusztig polynomials of matroids always unimodal - in particular log-concave, or even real-rooted, as conjectured? No: representable matroids obtained by deleting points from finite projective geometries have non-unimodal Kazhdan-Lusztig polynomials over every finite field, so the log-concavity and real-rootedness conjectures are both false.
Give an explicit profinite presentation of $\operatorname{Gal}(\overline{\mathbb{Q}}_2 / \mathbb{Q}_2)$. The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-$2$ condition on the wild generators.
Simon conjectured that every skeleton of a simplex is extendably shellable. False: for every $d \ge 3$ there is a pure $d$-dimensional shellable simplicial complex that is not shelling completable.
rules out the two-point LP method in this dimension; the optimal packing in dimension 36 remains unknown
The paper constructs an exact cluster $F\subseteq\mathbb{Z}^2$ of cardinality 8 with full affine span and an $F$-tiling whose orbit closure contains no 1-periodic $F$-tiling, giving a non-degenerate counterexample to Nivat's conjecture for non-convex windows. This answers negatively a question of Kari and Moutot from 2023.
Whether the real Kalton-Peck space $Z_2$ is isomorphic to its hyperplanes. It is not: no hyperplane of $Z_2$ is isomorphic to $Z_2$, proved through a rank parity theorem for symplectic spaces applied to the Kalton-Swanson symplectic structure.
Does the value of a two-player quantum game decay exponentially under parallel repetition, as Raz's theorem gives for classical games? Yes: an exponential parallel repetition theorem holds for arbitrary finite two-player quantum games.
For the integral $\mathcal{I}(h)$ over the unit interval and the torus, the paper gives the three-term Laurent polynomial $f(x,z)=(1-z^{-1})((1-x)+xz)$ with $\mathcal{I}(f^n)=0$ but $\mathcal{I}(z^{-1}f^n)=(-1)^{n-1}/(n+1)\neq0$. This disproves the $xz$-conjecture with one interval and one torus variable, shows $\ker\mathcal{I}$ is not a Mathieu–Zhao subspace, and by padding yields counterexamples for SU(2).
Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)
Charts the bounded-slope regime of the arithmetic Kakeya program; the conjecture itself remains open.
Fully resolves the posed coordinate-wise question; the main Courtade-Kumar conjecture itself remains open outside the extended high-noise range.
One finite-dimensional construction settles several related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time.…
The theorem improves the best lower bound valid in *every* sufficiently large genus from asymptotic constant $2/9$ to $1$. The every-genus ladder it climbs is Katz-Sabourau's $19/120$ and then Liu-Petri's $2/9$, the latter also by a random construction. Constant $1$ was already reached by Petri-Walker along a subsequence of genera, following Erdos-Sachs, so the new contribution is achieving it uniformly rather tha…
The counterexamples come with divisibility bounds on the Hodge-theoretic index, at 2-torsion and 5-torsion, on very general hyperkahler fourfolds.
The paper records how stuck this was: the first author had discussed the question with Caucher Birkar, Osamu Fujino, Christopher D. Hacon, Junpeng Jiao, Vladimir Lazic and Lingyao Xie, and writes that despite a general feeling that a negative answer was likely, no precise counterexample could be found. The authors also note that, given the limitations of generative AI, they may have missed related literature and w…
the equality case; the inequality was settled separately and is tracked on its own entry
Does every nonlocal game admitting a perfect entangled strategy admit one using a maximally entangled state? Described in the paper as one of the longstanding open problems in quantum nonlocality. Answered negatively by an explicit counterexample game.
The domain has dihedral symmetry of order 26 and is neither a disc nor centrally symmetric, and its eigenfunction changes sign - which is why an additional sign assumption rescues the statement.
Let $R(3;k)$ be the least $n$ such that every $k$-colouring of the edges of $K_n$ contains a monochromatic triangle. Determine $\lim_{k\to\infty} R(3;k)^{1/k}$ (a \$250 Erdős prize problem). A superexponential lower bound resolves the problem: the limit is infinite.
Amdeberhan, Shareshian and Stanley showed a function from the theory of partition Eisenstein series counts alternating permutations with a given record partition, and asked whether a similar theory exists for record compositions, suggesting a role for noncommutative symmetric functions. The paper solves that open problem with a product formula.
Furthest Pair and its relatives admit $f(d)\,n^{2-\Theta(1/d)}$ algorithms, making them the standard examples of barely subquadratic computation, and whether that is optimal in superconstant dimension was open. Under SETH it is: Furthest Pair requires quadratic time once the dimension is superconstant.